Linear Functional Equations and Convergence of Iterates Axel Torshage June 2012 Bachelor’sthesis 15 Credits Umeå University Supervisor - Yuriy Rogovchenko Abstract The subject of this work is functional equations with direction towards linear functional equations. The …rst part describes function sets where iterates of the functions converge to a …xed point. In the second part the convergence property is used to provide solutions to linear functional equations by de…ning solutions as in…nite sums. Furthermore, this work contains some transforms to linear form, examples of functions that belong to di¤erent classes and corresponding linear functional equations. We use Mathematica to generate solutions and solve itera- tively equations. Contents 1 Introduction 3 1.1 Historical perspective . 3 1.2 Idea .................................. 6 1.3 Notation................................ 7 1.3.1 Common notation . 7 1.3.2 Kuczma’snotation . 7 1.3.3 New classes of functions . 8 2 Convergence of function iterates 9 2.1 Counterexample . 11 2.2 Standardsets ............................. 11 2.3 Simple extension . 12 2.4 Extension of the function’sclass . 14 3 Linear Functional Equations 19 3.1 Homogenous linear functional equations . 19 3.1.1 Case (i) ............................ 21 3.1.2 Case (ii) ............................ 22 3.1.3 Case (iii) ........................... 23 3.2 Criteria for Gn ............................ 24 3.3 Particular solution . 26 3.3.1 Unique continuos solution . 26 3.3.2 Arbitrary constant solution . 30 3.3.3 Arbitrary functions solution . 32 4 Transformations 35 4.1 Transformation to simpler linear form . 35 4.2 Transformation of a general equation . 36 1 5 Examples 38 5.1 Convergence.............................. 38 0 5.1.1 A function from R [I] .................... 38 0 5.1.2 A function from S [I] .................... 39 0 5.1.3 A function from P [I] .................... 41 0 5.1.4 A function from T [I] .................... 42 5.2 Solutions to linear functional equations . 44 0 5.2.1 A function from R [I] .................... 44 0 5.2.2 A function from S [I] .................... 45 0 5.2.3 A function from P [I] .................... 46 0 5.2.4 A function from T [I] .................... 46 6 Conclusions 49 7 Appendix 51 1. Introduction In simple words, a functional equation is much like a regular algebraic equation, though instead of unknown elements in some set we are interested in …nding a function satisfying our equation. A typical example of a functional equation is ' (x) + ' x2 = x; where one has to …nd the function ' de…ned in some given interval. This equa- tion has a unique solution in [0; 1) and another one in (0; ) but no continuous solutions in [0; ) : In Section 3 we will see that this example1 is linear as well. 1 1.1. Historical perspective Even though functional equations were known long time before 17th century, it took another two hundred years until mathematicians …rst tried to organize the notations and theory. Grégoire de Saint-Vincent’s (1584-1667) work is a good starting point in the subject. He had noticed that the area under hyperbolic 1 graphs, for example, g (x) = x ; can be described by a function with the proper- ties ' (x)+' (y) = ' (xy). This is the property possessed by logarithmic functions which is easily shown by modern calculus, but in that time Grégoire used a geo- metric argument to obtain a functional equation, which would ultimately have a logarithmic solution. However, the subject of functional equations at that time was rather undevel- oped and we can date the real birth of the subject with the famous Augustin-Louis Cauchy (1789-1857). Despite being most famous for his work in mathematical analysis, Cauchy has provided the basis of functional equations as well. The Cauchy equation ' (x + y) = ' (x) + ' (y) is one of the most important functional equations and is the …rst property of a linear functional often used in analysis and linear algebra. Cauchy also gave name 3 Figure 1.1: Grégoire de Saint-Vincent (1584-1667) to the exponential counterpart ' (x + y) = ' (x) ' (y) and found the complete set of functions solving the d’Alambert equation ' (x + y) + ' (x y) = 2' (x) ' (y) : We introduce one more mathematician’swork. Charles Babbage (1791-1871) is known as a pioneer within computer science. Babbage’s work is closer to the direction this paper has due to the fact that he studied various functional equations with only one variable instead of the two used in the examples above. One family of functional equations that Babbage introduced is of the type G [x; ' (x) ;' (a1 (x)) ; ::::; ' (an (x))] = 0; where G; a1; ::::; an are known functions and ' is the unknown. This equation is of interest for us because by de…ning G in a certain way we can obtain a linear functional equation. ' (f (x)) g (x) ' (x) F (x) = 0: We conclude this part by giving some further contribution to Babbage. Before we Figure 1.2: Augustin-Louis Cauchy (1789-1857) Figure 1.3: Charles Babbage (1791-1871) x 15 10 5 x 4 2 2 4 5 10 15 Figure 1.4: Solutions of the Babbage equation. focus the linear functional equation, we shed some light on the Babbage equation. Though not linear, the following functional equation has some iterative values that we may …nd interesting in this work: ' (' (x)) x = 0: (1.1) Equation (1.1) is known as Babbage equation and involves an iteration of the unknown function ': The set of solutions to this equation is C C x ' (x) = x; ' (x) = C x; ' (x) = ;' ; x 1 + Dx where C and D are arbitrary real constants. Here we plot four solutions to equa- tion (1.1), corresponding to the values C = 7 and D = 4: 1.2. Idea The subject of functional equations has many similarities with that of di¤erential equations. Just as in the case of di¤erential equations the appearance of the functional equation is crucial for the solution method. One di¤erence is that for real-valued functional equation one uses function iterates to …nd a solution. Convergence of these functions is an important property that allows to …nd a solution by the methods described in this paper. In this project, we discuss only linear functional equations. 1.3. Notation We start by de…ning some notations that we will use in the sequel. Some of these notations might seem unrelated to the subject, but many of them describe quite important properties of functional equations. Much of the notation used in this paper coincides with the notation used in Marek Kuczma’smonograph [1], which is the main source of information in this work. 1.3.1. Common notation De…nition 1. We denote the most common interval as I; where I can be any real interval that …nite or in…nite. If nothing else is indicated, all functions are de…ned as f : I I; and x I respectively. When a function is said to converge, it is understood! that it converges2 in the interval I: De…nition 2. Cn [I] denotes the n times di¤erentiable functions in the interval I: In this work we will at most times only demand that the functions are continuous, meaning that f C0 [I]. 2 1.3.2. Kuczma’snotation De…nition 3. For a given f (x) and an interval I such that f : I I; we de…ne the n-th iterate as ! 0 n+1 n f (x) = x; f (x) = f (f (x)) ; x E; n N 0 : 2 2 [ f g Not to be confused with exponentiation, which will be denoted (f (x))n and like- wise sin2 (x) = sin (sin (x)) ; not (sin (x))2 : De…nition 4. A …xed point I corresponding to a certain function f is a point such that f () = : 2 There are certain sets that are of special interest, the intervals or sets where our functions are de…ned. We will also de…ne some subsets of the set of functions de…ned in I. De…nition 5. The set Sn [I] Cn [I] is the set of functions such that (f (x) x)( x) > 0; x = ; (1.2) 6 and (f (x) )( x) < 0; x = : (1.3) 6 n n De…nition 6. R [I] denotes the subset of functions in S [I] that are strictly increasing. De…nition 7. [I] represents all functions ' : I ; where is a given set (in ! this work, = R). 1.3.3. New classes of functions De…nition 8. P n [I] is the set of functions P n [I] Cn [I] satisfying (1.2) and (f (x) + x 2)( x) < 0; x = : (1.4) 6 De…nition 9. T n [I] Cn [I] is the set of functions with the property (1.2) and f (x) ((k + 1) kx) < 0; x < ; 8 (1.5) kf (x) ((k + 1) x) > 0; x > ; 8 where k is any constant larger then 0. 2. Convergence of function iterates Convergence to a …xed point is a valuable property of functions in this topic, many theorems will demand that some functions in the linear functional equation behave in a certain manner. This chapter will discuss for which sets of functions we can ensure this convergence property. First, we show that for the two set n n S [I] and R [I] de…ned in Section 1.3 have convergence to a …xed point. Then, n n we show that the same is true for the sets P [I] and T [I]. There exist larger sets containing converging functions, but these are not considered in this work. We will …rst prove an important lemma. Lemma 10.